System and method for real-time passive seismic event localization

ABSTRACT

A computer-implemented method for seismic event localization includes: generating, with at least one processor, a vectorized snapshot matrix representing wave propagation data at a series of snapshots in time for a subterranean formation; computing a reduced orthonormal column basis matrix based on the vectorized snapshot matrix; constructing a reduced order wave propagation model based on the reduced orthonormal column basis matrix; receiving seismic data collected from a plurality of receivers at the subterranean formation; generating a time-domain coefficient matrix based on back propagation of the received seismic data and the reduced order wave propagation model; reconstructing time-reversed wavefield data based on the time-domain coefficient vector; and generating signals for outputting wavefield or seismic event location information based on the time-reversed wavefield data.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.16/683,719, filed Nov. 14, 2019, which claims all benefit includingpriority to Canadian Patent Application No. 3,024,240, filed Nov. 15,2018, and entitled, “SYSTEM AND METHOD FOR REAL-TIME PASSIVE SEISMICEVENT LOCALIZATION”. Each of these are hereby incorporated by referencein their entireties.

FIELD

Embodiments of the present disclosure relate generally to the field ofseismology, and some embodiments particularly relate to systems andmethods for seismic event localization.

BACKGROUND

Event localization is an important task for seismic monitoring,including locating earthquakes in global seismology (Fehler, 2008) andmicroseismicity in hydraulic fracturing treatment (van der Baan et al,2013, Li and Van der Baan, 2016). Time-reversal extrapolation is apromising event localization technique using full waveform information(McMechan et al, 1985, Artman et al, 2010), which can also be calledbackward wavefield extrapolation. In traditional time-reversalextrapolation, receivers are treated as sources and the three-componentwavefields are time-reversed and directly propagated toward its sourcelocation using a two-way elastic wave modeling operator and a knownvelocity model (McMechan, 1983). This method could be more accuratecompared to traditional travel-time based methods since no picking of P-and S-wave first arrivals is required.

Location results may be badly affected by mispicking and inaccurate(Castellanos and Van der Baan, 2015, Li and Van der Baan, 2016). Li andVan der Baan (2016) introduce an improved time-reversal extrapolationscheme, in which both particle velocity and pressure are extrapolatedseparately followed by their combination according to the acousticrepresentation theorem.

SUMMARY

For the purpose of computational efficiency, the same type of data canbe extrapolated simultaneously. However, the above-noted methods arestill both impractical for application in continuous real-timemonitoring because solving a very large simulation system composed ofseveral spatially discretized elastic wave equations is computationallyintensive.

Traditional time-reversal extrapolation is a promising technique forpassive seismic event localization. However, the technique is based onsolving discrete two-way wave equations using the finite difference orfinite element method, which makes it very time-consuming and notsuitable for real-time applications. The generated wavefields haveinformation redundancy that only a small amount of information is enoughto represent the whole extrapolation process.

In some embodiments, proper orthogonal decomposition is used to removeinformation redundancy and create a much smaller extrapolation system,with which real-time or near real-time passive seismic eventlocalization is possible. Aspects of the present disclosure describe aextrapolation system which applies proper orthogonal decomposition tothe first-order two-way elastic wave equations. In some embodiments, theextrapolation system is used to build a continuous waveform basedpassive seismic event localization scheme that can, in some instances,rapidly locate multiple seismic events and determine their origin time.

In accordance with an aspect of the present disclosure, there isprovided a computer-implemented method for seismic event localization.The method includes: generating, with at least one processor, avectorized snapshot matrix representing wave propagation data at aseries of snapshots in time for a subterranean formation; computing areduced orthonormal column basis matrix based on the vectorized snapshotmatrix; constructing a reduced order wave propagation model based on thereduced orthonormal column basis matrix; receiving seismic datacollected from a plurality of receivers at the subterranean formation;generating a time-domain coefficient matrix based on back propagation ofthe received seismic data and the reduced order wave propagation model;reconstructing time-reversed wavefield data based on the time-domaincoefficient vector; and generating signals for outputting wavefield orseismic event location information based on the time-reversed wavefielddata.

In accordance with another aspect, there is provided a computing systemfor seismic event localization. The system includes: at least one memoryand at least one processor. The at least one processor is configuredfor: generating a vectorized snapshot matrix representing wavepropagation data at a series of snapshots in time for a subterraneanformation; computing a reduced orthonormal column basis matrix based onthe vectorized snapshot matrix; constructing a reduced order wavepropagation model based on the reduced orthonormal column basis matrix;receiving seismic data collected from a plurality of receivers at thesubterranean formation; generating a time-domain coefficient matrixbased on back propagation of the received seismic data and the reducedorder wave propagation model; reconstructing time-reversed wavefielddata based on the time-domain coefficient vector; and generating signalsfor outputting wavefield or seismic event location information based onthe time-reversed wavefield data.

In accordance with another aspect, there is provided a non-transitorycomputer-readable medium or media having stored thereoncomputer-readable instructions which when executed by at least oneprocessor, configure the at least one processor to perform at least someaspects of any of the methods described herein.

DESCRIPTION OF THE FIGURES

Reference will now be made to the drawings, which show, by way ofexample, embodiments of the present disclosure.

FIG. 1 shows relative locations of example wavefield components in astaggered grid.

FIG. 2 shows aspects of an example subterranean formation, wells andexample training and target areas.

FIG. 3 shows example Marmousi velocity models. (a) shows a Marmousivelocity model for microseismic monitoring simulation where the starsrepresent miscroseismic events; (b) shows a Smoother Marmousi velocitymodel for time-reversal extrapolation. The triangles represent receiversin a borehole.

FIG. 4 shows source time functions a) used for off-line training, and b)used for on-line simulation.

FIG. 5 shows a singular value plot of an example snapshot matrix A.

FIG. 6 shows visual representations of wavefield data.

FIG. 7 shows a comparison of waveforms calculated by reduced-order andhigh-fidelity simulations.

FIG. 8 shows particle velocities in the x direction (left panel) and zdirection (right panel).

FIG. 9 shows energy flux maps using high-fidelity reconstructions, andreduced-order systems.

FIG. 10 is a chart illustrating computation versus recording time.

FIG. 11 is a diagram showing aspects of an example system and processfor seismic event localization.

FIG. 12 outlines aspects of an example first or offline training stage.

FIG. 13 outlines aspects of an example second or online source imagingstage.

FIG. 14 outlines aspects of an example simulation phase of an examplefirst or offline training stage.

FIG. 15 shows an example data structure of an example simulation phase.

FIG. 16 outlines aspects of an example learning phase of an examplefirst or offline training stage.

FIG. 17 outlines aspects of an example reduced order model buildingphase of an example first or offline training stage.

FIG. 18 outlines aspects of an example coefficient calculation phase ofan example second or online localization stage.

FIG. 19 outlines aspects of an example location determination phase ofan example second or online localization stage.

FIG. 20 shows an example data structure of an example locationdetermination phase.

FIG. 21 shows aspects of an example method for seismic eventlocalization.

DESCRIPTION OF EXAMPLE EMBODIMENTS

FIG. 2 shows a cross-section of an example subterranean formation towhich aspects of the present disclosure may be applied. In the exampleformation, is a treatment well and an observation well. Three hydraulicfracturing stages are at the end of the treatment well. The trianglesrepresent receivers in a borehole. As described herein or otherwise, insome embodiments, the area surrounded by the outer box can represent amodel used for training, which the inner box can represent a smallertarget area in which wavefields can be reconstructed.

FIG. 11 shows a dataflow diagram showing multiple modules or processingstages, and the general direction of the flow of data. In someembodiments, the modules can be software code, hardware, or acombination thereof which include or configure one or more processingdevices to perform one or more aspects of the processes illustrated inFIG. 11 or described herein or otherwise.

In some embodiments, a system or device can include one or moreprocessors, one or more memories and/or data storage devices, one ormore communication interfaces, one or more input and/or output devices(e.g. displays), and/or one or more receivers at a subterraneanformation (e.g. within or at the surface of a subterranean formation).

In some embodiments, the processor(s) are configured to perform one ormore of the aspects of the methods and processes described herein.

In some embodiments, the processor(s) are configured to perform a methodfor seismic event localization. In some embodiments, as illustrated inFIG. 11, the method can include a first or offline training stage, and asecond or online stage.

As described herein or otherwise, in some embodiments, the first stagecan include simulation process(es), machine learning or trainingprocess(es), and process(es) for building a reduced order model.

As described herein or otherwise, in some embodiments, the second stagecan include process(es) for real-time coefficient computation, andwavefield reconstruction in a spatiotemporal domain.

Aspects of an example offline or first stage are outlined in FIGS. 12,14, 16 and 17. Aspects of an example online or second stage are outlinedin FIGS. 13, 18, and 19.

As described herein or otherwise, FIG. 21 shows aspects of an examplemethod for seismic event location. At 2110, one or more processor(s)generate a vectorized snapshot matrix representing wave propagation dataat a series of snapshots in time for a subterranean formation. In someembodiments, generating the vectorized snapshot matrix includessimulating wave propagations on a set of spatially discretized partialdifferential equations; and extracting simulated wave propagation dataat each snapshot in the series of snapshots in time.

In some embodiments, simulating the wave propagations uses sourcepositions corresponding to locations of physical receivers at thesubterranean formation. In some embodiments, locations at thesubterranean formation can include locations within the subterraneanlocation (e.g. locations in wells), locations around the subterraneanformation (e.g. at surface level), or any other location at whichreceivers may be positioned to receive relevancy seismic or other data.

In some embodiments, the set of spatially discretized partialdifferential equations include 2-dimensional stress-velocity two-wayelastic wave equations as described herein or otherwise.

In some embodiments, the wave propagation data is generated with avelocity model associated with the subterranean formation. In someembodiments, this is based on previously collected data or otherwiseknown, assumed, predicted or calculated parameters or characteristics ofthe subterranean formation.

In some embodiments, the vectorized snapshot matrix can be generated toinclude wave propagation data and absorbing layer values as illustratedfor example in FIG. 15.

At 2120, as described herein or otherwise, the processor(s) compute areduced orthonormal column basis matrix based on the vectorized snapshotmatrix. In some embodiments, computing the reduced orthonormal columnbasis matrix includes performing a singular value decomposition and/orQR decomposition.

In some embodiments, computing the reduced orthonormal column basismatrix is based on multi-dimensional stress-velocity two-way elasticwave equations as described herein or otherwise.

At 2130, as described herein or otherwise, the processor(s) construct areduced order wave propagation model based on the reduced orthonormalcolumn basis matrix. In some embodiments, the reduced order wavepropagation model can include one or more data structures including datarepresenting values or parameters for defining the model.

At 2140, as described herein or otherwise, the processor(s) receiveseismic data collected from a plurality of receivers at the subterraneanformation. As noted above, receivers at the subterranean formation caninclude receivers positioned within, above or otherwise around theformation. In some embodiments, seismic data can include pressure data,particle velocity data and/or any other data which can be used forseismic event localization.

At 2150, as described herein or otherwise, the processor(s) generate atime-domain coefficient matrix based on back propagation of the receivedseismic data and the reduced order wave propagation model.

At 2160, as described herein or otherwise, the processor(s) reconstructtime-reversed wavefield data based on the time-domain coefficientvector.

In some embodiments, reconstructing the time-reversed wavefield data isbased on reconstructing a target area of the subterranean formation. Asillustrated by the target area in FIG. 2, in some instances, the targetarea is a subset of or is otherwise smaller than the subterraneanformation. In some embodiments, reconstructing only the target area canbe faster than reconstructing the wavefield data across the entireformation. In some embodiments, the target area is a volume of thesubterranean formation.

At 2170, as described herein or otherwise, the processor(s) generatesignals for outputting wavefield or seismic event location informationbased on the time-reversed wavefield data. In some embodiments,outputting the wavefield or seismic event location information caninclude storing the information, generating visual representations (fordisplay on a screen or printout), generating visual or audible alerts,sending communications to recipients or other systems/devices, and thelike.

In some embodiments, the processors are configured to use properorthogonal decomposition (POD). In some embodiments, this can be basedon Pereyra and Kaelin (2008) which proposes a fast acoustic wavefieldpropagation simulation procedure by constructing an order-reducedmodeling operator, which provides a possible solution to the issue ofhigh computational cost mentioned above. In some instances, this mayspeed up simulation by projecting the spatially discretized waveequations from a higher dimensional system to a much lower dimensionalsystem, still keeping sufficient accuracy (Pereyra and Kaelin, 2008).

POD is a technique aimed at reducing the complexity of a numericalsimulation system using mathematical insights. In some embodiments, PODcan be applied in many dynamic system simulations as shown in(Chatterjee, 2000) and (Schilders, 2008). Applications include modelingof fluid flow, real-time control, heat conduction (Lucia et al., 2004),wavefield propagation (Pereyra and Kaelin, 2008, Wu et al., 2013),aircraft design (Lieu et al, 2006), arterial simulations (Lassila et al,2013) and nuclear reactor core design (Sartori et al, 2014), etc. POD isbased on the observation that simulations with a high computational loadoften repeatedly solve the same problem (Benner et al, 2015).

In the case of time-reversal extrapolation for microseismic eventlocalization, the continuously recorded data are back-propagated throughan unchanging velocity model. Thus the wave propagation ‘engine’ doesnot change, only the recorded data vary. Also, informational redundancyexists in most of the traditional simulation processes (Schilders,2008), which means the discretized wave equations can be represented asa large but sparse matrix which can be compressed into a small butdenser system that is much faster to solve.

In accordance with one aspect, the present disclosure describes anadaptive randomized QR decomposition (ARQRd) based POD method. In someinstances, this method may balance accuracy and computationalefficiency.

The basic idea is to project the original data in a high-dimensionalspace to a low-dimensional data space using randomly selected vectors.However, the dimension of the new space is usually unknown (Halko etal., 2010) and sometimes needs to be pre-tested before the new matrixcan capture the demanded amount of features in the original data. In thepresent disclosure, ARQRd may automatically provide the new projecteddata which not only have the minimum dimension but also capture the mostamount of information without the need for pretesting.

In some embodiments, a reduced-order two-way elastic wave modelingsystem is built that can be used for time-reversal extrapolation usingthe proposed POD process. In some embodiments, a POD-based energy fluxbased focusing criterion can be used to fit in the low-order modelingscheme (Li and Van der Baan (2017)). The new wavefield extrapolationapproach is more computationally efficient which makes real-timeautomatic seismic event localization possible. As illustrated herein,the ARQRd results have been compared with the results from traditionaltime-reversal extrapolation.

Similar techniques, such as randomized SVD and randomized QR, have beenapplied as rapid rank approximation methods for geophysical purposes(Gao et al., 2011, Oropeza and Sacchi, 2011, Cheng and Sacchi, 2015),especially when dealing with large datasets.

POD generally has two steps, namely an off-line training part where thesmaller simulation system is learned and created, and an on-linecalculation where the data are repeatedly generated with little cost andhigh accuracy.

1.1 Off-Line Training

In some embodiments, off-line training is an aspect of POD whichincludes the following steps:

(1) Compute training data and construct a snapshot matrix, which isformed from high-fidelity simulations. In some embodiments, ahigh-fidelity simulation is calculated by numerically solving a group ofspatially discretized partial differential equations (PDEs), given by

$\begin{matrix}{{\frac{\partial{u(t)}}{\partial t} = {{L{u(t)}} + {f(t)}}},} & (1)\end{matrix}$

where u is a time dependent state variable, which is spatially discretebut continuous in time; L is a matrix of partial differential operators,f is a time dependent source term, the total discretized simulation timeis N_(T). Equation 1 is a general form of discretized PDEs that isapplicable in 1D, 2D or 3D wave simulations. Finite difference or finiteelement methods are the two most common methods to solve equation 1.Since equation 1 represents a time-varying system, a snapshot means onetime slice of a high-fidelity simulation of the system. A series ofsnapshots extracted during the full simulation are put into snapshotmatrix A_(s) _(i) in chronological order, given by

A _(s) _(i) =[u ₁ ^(i) ,u ₂ ^(i) ,u ₃ ^(i) , . . . ,u _(N) _(t)^(i)],  (2)

where s_(i) means the i^(th) simulation; each column u_(t) ^(i)corresponds to a vectorized snapshot at time t and the subscript indexN_(t) is the total number of time slices used for training, whereN_(t)≤N_(T) and the sampling time interval between any two adjacentslices has to satisfy the Nyquist sampling theorem. For multiplesimulations, an even larger snapshot matrix A is constructed comprisingmultiple snapshot matrix, given by

A=[A _(s) ₁ ,A _(s) ₂ ,A _(s) ₃ , . . . ,A _(s) _(n) ],  (3)

where n is the number of simulations. Each simulation s_(i) canrepresent different aspects, for instance, different source positions ordifferent source radiation patterns.

(2) Compute and compress a left orthogonal basis of snapshot matrix A.The basic assumption of POD is that matrix A can be approximated by anew matrix Q whose rank is much lower than the size of either dimensionof matrix A, while still keeping most of the key information of A,denoted by

Rank(Q)<<Minimum(m,n),  (4)

where m and n are the row and column numbers of snapshot matrix Arespectively. The selection of Q is non-unique. In some embodiments, Qis defined as an orthonormal basis such that each column of the matrix Acan be expressed by a combination of columns in Q with sufficientaccuracy. Since Q is an orthonormal matrix, it also satisfies thecondition

Q ^(T) Q=I,  (5)

where I is an identity matrix (Strang, 2006).

In some embodiments, singular value decomposition (SVD) or QRdecomposition (QRd) can be used to compute the left orthonormal basis Qand the singular values of the snapshot matrix A. The approximated rankR_(A) is determined by choosing the largest singular values and thecorresponding columns are grouped into Q which is only a small portionof the full left orthonormal basis, as long as the selected basis Q isenough to span the column space of the snapshots matrix A,mathematically satisfying the evaluation criterion

∥A−QQ ^(T) A∥≤ε  (6)

where ∥⋅∥ denotes the l₂ norm and ε is a positive error tolerance (Halkoet al., 2010). The selected basis Q can be referred to as the reducedorthonormal column basis.

In some embodiments, ARQRd is used to calculate the left orthonormalbasis Q. It can be seen as a randomized Gram-Schmidt method embeddedwith the evaluation criterion (equation 6), where the reducedorthonormal column basis Q of the snapshot matrix A is calculated in aniterative scheme (Halko et al., 2010).

In the i^(th) iteration, a new column vector c_(i) is first calculatedthrough a projection of snapshot matrix A using

c _(i) =Aω _(i),  (7)

where ω_(i) is a random column vector with a Gaussian distribution. Thenc_(i) is orthonormalized to all previously generated i−1 columns usingthe Gram-Schmidt method before it is added to the desired basis Q.Equation 6 is evaluated in each iteration so that the calculated basis Qhas a minimum number of columns satisfying the evaluation criterion witha given error tolerance E, when iteration stops. This makes ARQRd morecomputationally efficient than traditional randomized QR or SVD (Halkoet al., 2010).

(3) Construct a new reduced simulation system. In some embodiments, thenew reduced system approximates the full system (1), as long as thesource positions are unchanged. The source waveforms can be differentbut must have an overlapping frequency content. The state variable u(t)can then be expressed as a linear combination of the reduced orthonormalcolumn basis Q, using

u(t)=Qa(t),  (8)

where a(t) is a coefficient vector at time t. Note equation 8 is the keyassumption that makes this method successful. Equation 8 is substitutedinto equation 1 giving

$\begin{matrix}{{\frac{{\partial Q}{a(t)}}{\partial t} = {{LQ{a(t)}} + {f(t)}}}.} & (9)\end{matrix}$

Given that matrix Q is time independent, we can further rephrase theequation by multiplying its both sides with Q^(T), rendering

$\begin{matrix}{{\frac{\partial{a(t)}}{\partial t} = {{Q^{T}LQ{a(t)}} + {Q^{T}{f(t)}}}},} & (10)\end{matrix}$

where Q^(T)LQ is a reduced-order partial differential operator matrix ofsize [N_(Q),N_(Q)]; Q^(T)f(t) a reduced-order source term. It can beseen from equation 4 that the dimensions of the new simulation system ismuch smaller than the original one (equation 1), which ensures therepeated simulations can be done on-the-fly.

1.2 On-Line Simulation

In some embodiments, Equation 10 is solved on-the-fly using a finitedifference method, where both spatial and temporal axes are discretized,given by

a _(i+1) −a _(i−1) =La _(i) +Q ^(T) f _(i),  (11)

where L is the reduced finite difference operator Q^(T)LQ scaled by dt;a_(i+1) and a_(i−1) are coefficient vectors at discrete time point i+1,i and i−1; Q is the basis matrix scaled by dt.

At each time iteration, the coefficient vector a is updated usingequation 11 and saved for wavefield construction. Then, any snapshotu_(i) at time point i can be reconstructed using equation 8.

2 Reduced-Order Time-Reversal Extrapolation

Two dimensional (2D) stress-velocity two-way elastic wave equations aregiven by

$\begin{matrix}{{\frac{\partial{v_{x}(t)}}{\partial t} = {{\frac{1}{\rho}\frac{\partial{\tau_{xx}(t)}}{\partial x}} + {\frac{1}{\rho}\frac{\partial{\tau_{xz}(t)}}{\partial z}} + {f_{v_{x}}(t)}}}{\frac{\partial{v_{z}(t)}}{\partial t} = {{\frac{1}{\rho}\frac{\partial{\tau_{xz}(t)}}{\partial x}} + {\frac{1}{\rho}\frac{\partial{\tau_{zz}(t)}}{\partial z}} + {f_{v_{z}}(t)}}}{\frac{\partial{\tau_{xx}(t)}}{\partial t} = {{\left( {\lambda + {2\mu}} \right)\frac{\partial{v_{z}(t)}}{\partial x}} + {\lambda\frac{\partial{v_{z}(t)}}{\partial z}} + {f_{p}(t)}}}{\frac{\partial{\tau_{zz}(t)}}{\partial t} = {{\left( {\lambda + {2\mu}} \right)\frac{\partial{v_{z}(t)}}{\partial z}} + {\lambda\frac{\partial{v_{x}(t)}}{\partial x}} + {f_{p}(t)}}}{{\frac{\partial\tau_{x{z{(t)}}}}{\partial t} = {\mu\left( {\frac{\partial{v_{x}(t)}}{\partial z} + \frac{\partial{v_{z}(t)}}{\partial x}} \right)}},}} & (12)\end{matrix}$

where τ_(xx) and τ_(zz) are the x and z components of the normal stressfields; τ_(xz) is shear stress field; v_(x) and v_(z) are horizontal andvertical components of particle velocity fields; f_(v) _(x) and f_(v)_(z) are single forces and f_(p) is an external pressure source.

Equation 12 is written into the matrix form of equation 1 by assigningthat

$\begin{matrix}{{u = \begin{pmatrix}v_{x} \\v_{z} \\\tau_{xx} \\\tau_{zz} \\\tau_{xz}\end{pmatrix}},{L = \begin{pmatrix}0 & 0 & {\frac{1}{\rho}L_{x}} & 0 & {\frac{1}{\rho}L_{z}} \\0 & 0 & 0 & {\frac{1}{\rho}L_{z}} & {\frac{1}{\rho}L_{x}} \\{\left( {\lambda + {2\mu}} \right)L_{x}} & {\lambda L_{z}} & 0 & 0 & 0 \\{\lambda L_{x}} & {\left( {\lambda + {2\mu}} \right)L_{z}} & 0 & 0 & 0 \\{\mu\; L_{z}} & {\lambda L_{x}} & 0 & 0 & 0\end{pmatrix}},{f = \begin{pmatrix}f_{v_{x}} \\f_{v_{z}} \\f_{p} \\f_{p} \\0\end{pmatrix}},} & (13)\end{matrix}$

where L_(x), L_(y) and L_(z) represent the matrix form of the spatialderivatives

$\frac{\partial}{\partial x},{{\frac{\partial}{\partial y}\mspace{14mu}{and}{\mspace{11mu}\;}\frac{\partial}{\partial z}};}$

u and f are the vectorized wavefields and source term respectively.

In some embodiments, a staggered grid finite difference method is usedto discretize the 2D model, where wavefield variables v_(x), v_(z),τ_(xx), τ_(zz) and τ_(xz) are assigned to the grid according to FIG. 1(Zeng and Liu, 2001), with the corresponding grid size [N_(z),N_(x)−1],[N_(z)−1,N_(x)], [N_(z),N_(x)], [N_(z),N_(x)] and [N_(z)−1,N_(x)−1].Since u is a vector composed of five vectorized wavefield variables, itslength is the sum of their total grid numbers, denoted by N_(m).Likewise, L is a sparse matrix with the size of [N_(m), N_(m)].

Next, the processor(s) are configured to build a reduced-order system oftwo-way elastic wave equations for time reversal extrapolation. Thoughthe derivation is in the 2D space for simplicity, it can be extended tothe 3D space with no problem. Moreover, two assumptions are needed forthis derivation, namely (1) receiver locations and approximated velocitymodel are known; (2) the same finite differential operator is used inboth forward and time-reversal extrapolations.

The idea of traditional time-reversal extrapolation is basically thesame as wave propagation simulation using equation 12. The onlydifference is that the source term f_(v) _(x) and f_(v) _(z) inequations 1 and 13 are replaced with the time-reversed x- andz-component of particle velocity recordings, which means in this methodreceivers are turned into sources from where recordings are injectedinto the medium (McMechan, 1983).

2.1 Offline Training

According to the previous description, the size of the snapshot matrix Ais directly determined by the number of receivers. For generality, weassume there are N_(s) receivers located at coordinates [x₁,z₁],[x₂,z₂], . . . , [x_(N) _(s) ,z_(N) _(s) ] for passive seismicmonitoring, where x and z are horizontal position and depthrespectively. Since in 2D reverse-time extrapolation, there arehorizontal and vertical components of recordings to be extrapolated,simulations from sources with a single component, horizontal andvertical, are both needed to generate both P- and S-wavefields, and allof these are included in one snapshot matrix A for training (Equation3). The snapshot matrix for a j^(th) horizontal single-force sources_(j) located at [x_(j),z_(j)] is

U _(s) _(j) ^(x)=[u ₁ ^(x) ,u ₂ ^(x) , . . . u _(N) _(t) ^(x)],  (14)

where superscript x refers to the horizontal direction of a single-forcesource; N_(t″)N_(T), where N_(T) is discrete total simulation time. Asimilar expression holds for U^(z) where superscript z denotes verticaldirection. Then the complete snapshot matrix for training is grouped as

$\begin{matrix}{A = \left\lbrack {U_{s_{1}}^{x},U_{s_{2}}^{x},{\ldots\mspace{14mu} U_{s_{N_{s}}}^{x}},U_{s_{1}}^{z},U_{s_{2}}^{z},{\ldots\mspace{14mu} U_{s_{N_{s}}}^{z}}} \right\rbrack} & (15)\end{matrix}$

Then we apply ARQRd to the snapshot matrix A to get a basis Q with thegrid size of [N_(m), N_(Q)] following the step (2) in the last section.Likewise, we construct a reduced-order partial differential operatorQ^(T)LQ, which can be used in time-reversal extrapolation for real-timepassive seismic localization. The size of the new partial differentialoperator is [N_(Q), N_(Q)]. Since N_(Q)<<N_(m), the size of thereduced-order extrapolation system is much smaller than the originalone.

Based on the previous derivation, the reduced-order time-reversalextrapolation system can be built by first replacing the source term fin equation 13 with the time-reversed horizontal and vertical componentsof particle velocity recordings R_(x) ^(tr) and R_(z) ^(tr)respectively, given by

$\begin{matrix}{{D^{tr} = \begin{pmatrix}R_{x}^{tr} \\R_{z}^{tr} \\0 \\0 \\0\end{pmatrix}},} & (16)\end{matrix}$

where the total temporal sampling number of recordings is N_(D).Equation 1 becomes

$\begin{matrix}{{\frac{\partial u^{tr}}{\partial t} = {{Lu^{tr}} + D^{tr}}},} & (17)\end{matrix}$

where u^(tr) are the time-reversed wavefields, denoted by

$\begin{matrix}{{u^{tr} = \begin{pmatrix}v_{x}^{tr} \\v_{z}^{tr} \\\tau_{xx}^{tr} \\\tau_{zz}^{tr} \\\tau_{xz}^{t\tau}\end{pmatrix}},} & (18)\end{matrix}$

Analogous to the derivation of equation 10, equation 17 is written intoa reduced-order form

$\begin{matrix}{{\frac{\partial{a^{tr}(t)}}{\partial t} = {{Q^{T}LQ{a^{tr}(t)}} + {Q^{T}D^{tr}}}},} & (19)\end{matrix}$

where a^(tr)(t) is the coefficient vector at time t for time-reversalextrapolation; Q^(T)D^(tr) are the reduced-order recordings as a sourceterm. Equation 19 is the reduced-order equation for time-reversalextrapolation.

2.2 Continuous Online Time-Reversal Extrapolation

With the previously derived reduced-order system, the processor(s) areconfigured to implement continuous online time-reversal extrapolation.In this step, since the total discrete simulation time in the offlinetraining step is N_(T), which is likely substantially less than thetotal sampling number of recordings N_(D), a discrete temporal window oflength N_(T) is used to select the reduced-order recording segments forextrapolation, using equation 11 and 19.

Then similar to equation 8, an equation

U ^(tr) =Qa ^(tr)  (20)

is used to reconstruct the complete time-reversed wavefields, where thestructure of the basis Q is

$\begin{matrix}{{Q = \begin{pmatrix}Q_{v_{x}} \\Q_{v_{z}} \\Q_{\tau_{xx}} \\Q_{\tau_{zz}} \\Q_{\tau_{xz}}\end{pmatrix}}.} & (21)\end{matrix}$

However, sometimes it is only necessary to reconstruct the wavefieldswithin a target area, which means only the portion of the basiscorresponding to the area is used in wavefield reconstruction, leadingto

u ^(tr) =Q _(new) a ^(tr),  (22)

where Q_(new) represents the portion of basis Q we used forreconstruction.

This describes the procedure of reduced-order extrapolation for a singlesegment of data, whereas for continuous extrapolation, a parameter N_(T)_(step) is used to move the temporal window of time-reversed data D^(tr)to the next segment, followed by the same extrapolation process.

During back-propagation, a source focusing criterion is needed due tothe absence of the zero-lag cross-correlation imaging condition which isnormally applied in time-reverse extrapolation based source localizationmethods (Artman et al, 2010). An energy flux based focusing criterioncan be applied to each time slice of the back-propagated source image toautomatically determine the source location based on the Houghtransform. Full details can be found in Li and Van der Baan (2016).

3 Example Embodiments

In some embodiments, the reduced-order system can be applied to twoexample applications, namely wavefield extrapolation and continuousmicroseismic event localization. Both examples use the Marmousi velocitymodel (FIG. 3), which is based on a profile through the North Quenguelatrough in the Cuanza basin (Mora, 2002). The model is 2928 m in depthand 9216 m in length. The discrete velocity model we used in thissection has grid numbers of 122 in depth and 384 in length, with a gridspacing of 24 m and time interval of 1.8 ms. The size of the finitedifferential operator (L in equation 11) for the high-fidelitysimulation is [233229, 233229], where 233229 is the number of rows of uin equation 13.

In both examples, the high-fidelity simulations are conducted by solvingthe traditional two-way wave equations using a staggered-grid finitedifference method, in which a fourth-order spatial and a second-ordertemporal finite difference operator are applied.

3.3 Reduced-Order Wavefield Extrapolation

This example embodiment is used to illustrate that input source timefunctions used for the on-line simulation can be different from the oneused for off-line training step. For simplicity, in an illustratedexample, only one source is used and is denoted by the second star fromthe top in image (a) of FIG. 3 and the smoothed Marmousi velocity model(image (b) of FIG. 3).

In the off-line training step, an explosive source with a Ricker waveletwith a peak frequency of 10 Hz is used, which originates at 0.01 s (FIG.4, graph (a)). The source locates on the right of the well withcoordinates (x, z) of (1900, 5500). The total simulation time is 1.5 s,with a temporal interval of 1.8 ms. We save every snapshot of bothparticle velocity and stress wavefields obtained from the high-fidelitysimulation to a snapshot matrix A according to equation 13 and 15. Thesize of A is [233229, 834], where 834 is the total number of discretetimes N_(t).

In order to illustrate the information redundancy of matrix A, wedisplay the singular values of A in FIG. 5, obtained by applying SVD toA. FIG. 5 shows a sharp drop in the singular values, showing informationredundancy exists in the traditional simulation process. We then applyARQRd to automatically calculate basis Q of the snapshot matrix A, wherethe value of ε is 10⁻⁶. The size of the basis Q is [233229, 216]. Thereduced-order extrapolation system is constructed according to equation10, in which the size of the reduced-order partial differential operatoris [216, 216]. The size of the new system is about 2×10⁻⁵ times of thatof the original extrapolation system.

In the online simulation step, a new source time function (FIG. 4, graph(b)) is applied as force f(t) in equation 10, using the reduced-ordermatrix Q as obtained from the simpler source, as shown in FIG. 4,graph(a). Wavefields are calculated using equations 8 and 10. As acomparison, also calculated are the wavefields from the new source timefunction (FIG. 4, graph (b)) using the high-fidelity simulation. Theresulting wavefields from the two simulation systems are shown in FIG.6.

The pressure wavefields in FIG. 6 at 0.71 s and 0.99 s are calculatedusing the reduced-order (FIG. 6, wavefields (a) and (b)) and thehigh-fidelity simulations (FIG. 6, wavefields (c) and (d)). FIG. 6,wavefields (e) and (f) show that the average differences between thepressure wavefields constructed by the two simulation schemes deviateless than 0.1% of the largest negative amplitudes. FIG. 6 shows a traceextracted from an arbitrary spatial location to compare the simulatedwaveforms in detail. The waveforms of pressure and x- and z-component ofparticle velocities from two simulation schemes overlap each otherperfectly. The above results show that the reduced-order simulation isinsensitive to changing source time functions in the on-line simulationstep as long as the frequency contents overlap and the source positionremains fixed. Wavefields derived from the reduced-order simulation arenear-identical to the ones obtained from the high-fidelity simulation.

Next, compared are the computational costs of 1.5 s of bothhigh-fidelity and reduced-order simulations, where the latter includesthe costs of the offline training, online calculation of coefficientsand wavefield construction, displayed in Table 1. For a fair comparison,the complete wavefields are calculated including two-component particlevelocities and normal and shear stresses in both cases. The totalcomputation time for the reduced-order simulation is 575.73 s, which ismuch longer than the cost of the high-fidelity simulation, 190 s.However, approximately 99% of computation costs are due to offlinetraining in order to obtain an order-reduced simulation system whereasthe calculation cost of coefficients only takes 0.04 of the totalcomputational cost. These results indicate suitability of time-reversalextrapolation for continuous passive seismic event localization, sincethe system/process is only needed to do the offline training once usinga limited total simulation duration whose computational cost is fixedand then it can be used repeatedly to extrapolate various recordingswith extremely fast speed, which eventually takes less computationaltime than high-fidelity simulations for longer recording time.

Also the offline training can further be sped up by reducing the numberof time snapshots in matrix A, equation 2 and 3 at the expense of lessaccurate reconstructions. For instance by including only one out ofevery three consecutive snapshots in time we obtain a much smallersnapshot matrix A. The offline training time becomes 480 s instead of575.73 s and the maximum reconstruction errors are less than 2% (FIGS.??i and ??j) for the same test setup. This is permissible as long as thedown-sampled snapshots matrix still actually reflects the frequencycontent of the complete data.

TABLE 1 Computation costs of 1.5 s high-fidelity and reduced-ordersimulations High fidelity Total cost 190 s Reduced order Offlinetraining 574 s Online calculation of coefficients 0.03 s Wavefieldconstruction 1.7 s

3.4 Continuous Microseismic Event Localization

A microseismic monitoring setup in this example is simulated, as shownin FIG. 3, image (a). In the model, a vertical borehole is simulatedwith four receivers at depths approximately from 1000 m to 2900 m torecord acoustic emissions from three double-couple sources. Particlevelocities in the x and z direction are measured at each receiver, witha total recording time of 9 s (FIG. 8).

FIG. 3, image (b) shows a smoothed Marmousi velocity model used foroff-line training and wavefield extrapolation. The model has the samediscrete size as the non-smoothed model. The smoothed velocity model isused to mimic the usual case in which a true velocity model is often notavailable and also to prevent secondary reflections.

Since x and z-component recordings of four receivers are to beextrapolated, eight sources corresponding to each component of the fourreceivers are used for simulations in off-line training step. They areall single force source, four in the x direction and four in the zdirection and all have a simulation time of 2 s. Ricker wavelets withpeak frequencies of 10 Hz are used in offline training. The wavefieldscorresponding to the eight sources are calculated separately. To reducememory issues in the ARQRd procedure, we only save every other snapshotin time obtained from each simulation to A using the ordering shown inequation 15. After applying ARQRd to matrix A, a basis Q is obtainedincluding all information of the wavefields radiated from the eightsources. The size of basis Q is [233229, 779]. The size of the newsystem is about 10⁻⁵ times of that of the original extrapolation system,whereas if only counting the non-zero elements of equation 1, the sizeof the new system becomes 0.4 times that of the original extrapolationsystem.

In the on-line extrapolation step, data are first segmented with a 2 stemporal window N_(T), denoted by AB in FIG. 8. The window is slidingalong the time axis in the direction of the black arrow in FIG. 8, witha sliding step N_(T) _(step) of 0.5 s. 0.5 s was chosen simply tobalance the number of events detected and total online extrapolationcost as for simplicity it was assumed there can be at most a singlemicroseismic event in each data segment. So with the time window slidingalong the recordings, the 9 s data are divided into fourteen segments.Each segment is injected into the reduced system respectively tocalculate coefficients a. Full wavefields are reconstructed usingequation 8. Only wavefields within a predefined area denoted by the redboxes in FIG. 9 were reconstructed, assuming that microseismic eventssolely occur here. All maps in FIG. 9 are absolute energy flux mapsusing the focusing criterion described in Li and Van der Baan (2017),for a better illustration.

Table 2 shows a comparison of the computation times in three scenarios,namely 1) direct back-propagation of 9 s recordings continuously usinghigh-fidelity simulation system; 2) direct back-propagation of fourteen2 s segments of recordings using high-fidelity simulation system; 3)offline training and back-propagation of fourteen 2 s segments ofrecordings using reduced-order system. It can be sees that reduced-ordersystem based time-reversal extrapolation of 9 s recordings takes moretotal computational time than direct extrapolation using high-fidelitysimulation system. But the computational costs of calculation ofcoefficients and reconstruction increase much slower than that whenusing high-fidelity system. In some embodiments, one may expect a muchless computational cost if much longer recordings are processed.

TABLE 2 Computation costs of continuous high-fidelity and reduced-ordersimulations Continuous 9s recordings High-fidelity Total cost 1140 s 14overlapping 2 s High-fidelity Total cost 3546 s segments (total 9 s) 14overlapping 2 s Reduced order Offline training 4146 s segments (total 9s) Online calculation of coefficients 0.36 s Wavefield construction 28 s

FIG. 9, snapshots a, b and c display the three snapshots of normalizedenergy flux maps using the high-fidelity reconstructions for the threefocusing maxima whose coordinates (x, z) are (2042,4950) (1898,5506) and(2100,6007) in meters. The locations of the maxima on each snapshot arealmost the same as the locations of the three predefined sources (bluestar in FIG. 9), whose coordinates (x, z) are (2040,4960) (1900,5500)and (2100,6000) in meters. Those side lobes on the maps are due to thevery limited number of receivers. The corresponding time of these threesnapshots are 1.1 s, 2.75 s and 5.68 s, which are close to the trueorigin times of 1.1 s, 2.8 s and 5.6 s. The high-fidelityreconstructions are compared with the ones obtained using thereduced-order systems (FIGS. 9 d, e and f) and find that the two resultsare essentially identical. This indicates that it is possible to docontinuous microseismic monitoring using the reduced-order systemreal-time and obtain similar results as for the high-fidelity system butwith substantially reduced online computation times.

In some instances, the devices/systems and methods using properorthogonal decomposition can be effective tools for creating asubstantially reduced simulation system by removing redundantinformation which normally exists in traditional two-way wave equationbased simulations. The reduced simulation system is significantly fasterwith good reconstruction quality. However, this may come at the cost ofa computationally intensive offline training step, which could be evenmore expensive than direct high-fidelity simulations. Generally, thecost of offline training is determined by the calculations of snapshotmatrix A and its left orthonormal basis Q, where the size of A isdirectly determined by the numbers of both high-fidelity simulationscorresponding to the included different sources and time slices selectedfrom each simulation for training. A snapshot matrix A is called acomplete snapshot matrix when it includes high-fidelity simulations withsources at every grid point within the model. Yet this may not berequired for all applications. For instance, a sufficient snapshotmatrix for time-reversal extrapolation only includes those simulationswith source locations corresponding to receiver locations in the realworld, which is a small portion of a complete snapshot matrix A.Conversely for fast forward modeling of waveforms due to sources at anypossible position (Pereyra and Kaelin, 2008), a near-complete snapshotmatrix is required. Whether it is necessary to apply the proposed methodto a certain application depends on the online versus offlinecomputation times as well as the computational resources available inthe online stage.

Two characteristics of microseismic monitoring permit and encourage thecreation of a reduced-order time-reversal extrapolation for real-timemicroseismic event localization, namely a limited number of receiversand long monitoring/recording time (usually from several hours to days).Borehole acquisitions typically use up to a dozen geophones, whereassurface acquisitions can be substantially larger ((Duncan and Eisner,2010; van der Baan et al, 2013). Yet it is not required to simulate asource at every possible spatial position in depth, greatly reducing thenumber of simulations which eventually leads to a more interestingsnapshot matrix A. Combined with the long recording times, this ensuresthat the overall computational time of reduced-order time-reversalextrapolation, including both offline and online calculation, is muchsmaller than the time required when using a high-fidelity simulationsystem. FIG. 10 shows a qualitative sketch of computation versusrecording time. The starting computation time of reduced-orderextrapolation (dash-dot line) is not zero because of offline training(dashed line) which could be more than direct high-fidelityextrapolation (solid line). The added computation time per reduced-ordersimulation is substantially smaller than that for the high-fidelityones. Hence at some recording length the two approaches use the sametotal computation times and reduced-order extrapolation uses lesscomputation time if recording time further increases.

To obtain both good performance and reasonable computation time, severalissues need to be addressed during the implementation of the proposedmethod for continuous time-reversal extrapolation. First, it is normallynot necessary to include every time slice obtained from high fidelitysimulations in snapshot matrix A as long as the time interval betweenselected two adjacent time slices satisfies Nyquist sampling theorem.Second, the computation of the left orthonormal basis Q is controlled bythe positive error tolerance ε. With a lower error tolerance ε, the leftorthonormal basis Q creates a more accurate but larger reduced-ordersystem since it captures more information in snapshot matrix A, whereasconversely, a higher error tolerance leads to a less accurate butsmaller reduced-order system. Finally, real data should be divided intosegments where the total time of each segment for online extrapolationis not longer than the simulation time T_(H) for offline training.Because no wavefield information at time over T_(H) is included neitherin snapshot matrix A nor in the reduced-order system, the computation ofcoefficient vector a(t) becomes unstable when extrapolation time islonger than T_(H).

Traditional simulation/extrapolation based on the two-wave wave equationis a high-fidelity but time-consuming process which has substantialinformation redundancy because discrete wavefields are similar withinadjacent spatial grids and temporal slices. It also repeatedly solvesthe same simulation problem since only the recorded data change but thevelocity field remains constant. In some embodiments, systems, devicesand methods described by way of example herein using proper orthogonaldecomposition can provide a technique to turn the high-fidelitysimulation into a much smaller system by removing the redundancy, whichcan be used to build a fast time-reversal extrapolation scheme. In someinstances, this may permit real-time waveform-based microseismic eventlocalization using feasible computational resources in the field.

Although the embodiments have been described in detail, it should beunderstood that various changes, substitutions and alterations can bemade herein without departing from the scope. Moreover, the scope of thepresent application is not intended to be limited to the particularembodiments of the process, machine, manufacture, composition of matter,means, methods and steps described in the specification.

As one of ordinary skill in the art will readily appreciate from thedisclosure, processes, machines, manufacture, compositions of matter,means, methods, or steps, presently existing or later to be developed,that perform substantially the same function or achieve substantiallythe same result as the corresponding embodiments described herein may beutilized. Accordingly, the appended claims are intended to includewithin their scope such processes, machines, manufacture, compositionsof matter, means, methods, or steps.

As can be understood, the examples described above and illustrated areintended to be exemplary only.

REFERENCES

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What is claimed is:
 1. A computer-implemented method for seismic eventlocalization, the method comprising: generating, with at least oneprocessor, a vectorized snapshot matrix representing wave propagationdata at a series of snapshots in time for a subterranean formation;computing a reduced orthonormal column basis matrix based on thevectorized snapshot matrix; constructing a reduced order wavepropagation model based on the reduced orthonormal column basis matrix;receiving seismic data collected from a plurality of receivers at thesubterranean formation; generating a time-domain coefficient matrixbased on back propagation of the received seismic data and the reducedorder wave propagation model; reconstructing time-reversed wavefielddata based on the time-domain coefficient vector; and generating signalsfor outputting wavefield or seismic event location information based onthe time-reversed wavefield data.
 2. The method of claim 1, whereincomputing the reduced orthonormal column basis matrix comprises:singular value decomposition or QR decomposition.
 3. The method of claim1, wherein computing the reduced orthonormal column basis matrix isbased on multi-dimensional stress-velocity two-way elastic waveequations:$\frac{\partial{v_{x}(t)}}{\partial t} = {{\frac{1}{\rho}\frac{\partial{\tau_{xx}(t)}}{\partial x}} + {\frac{1}{\rho}\frac{\partial{\tau_{xz}(t)}}{\partial z}} + {f_{v_{x}}(t)}}$$\frac{\partial{v_{z}(t)}}{\partial t} = {{\frac{1}{\rho}\frac{\partial{\tau_{xz}(t)}}{\partial x}} + {\frac{1}{\rho}\frac{\partial{\tau_{zz}(t)}}{\partial z}} + {f_{v_{z}}(t)}}$$\frac{\partial{\tau_{xx}(t)}}{\partial t} = {{\left( {\lambda + {2\mu}} \right)\frac{\partial{v_{z}(t)}}{\partial x}} + {\lambda\frac{\partial{v_{z}(t)}}{\partial z}} + {f_{p}(t)}}$$\frac{\partial{\tau_{zz}(t)}}{\partial t} = {{\left( {\lambda + {2\mu}} \right)\frac{\partial{v_{z}(t)}}{\partial z}} + {\lambda\frac{\partial{v_{x}(t)}}{\partial x}} + {f_{p}(t)}}$${\frac{\partial\tau_{x{z{(t)}}}}{\partial t} = {\mu\left( {\frac{\partial{v_{x}(t)}}{\partial z} + \frac{\partial{v_{z}(t)}}{\partial x}} \right)}},$where τ_(xx) and τ_(zz) are the x and z components of the normal stressfields; τ_(xz) is shear stress field; v_(x) and v_(z) are horizontal andvertical components of particle velocity fields; f_(v) _(x) and f_(v)_(z) are single forces and f_(p) is an external pressure source
 4. Themethod of claim 1, wherein generating the vectorized snapshot matrixcomprises: simulating wave propagations on a set of spatiallydiscretized partial differential equations; and extracting simulatedwave propagation data at each snapshot in the series of snapshots intime.
 5. The method of claim 4, wherein simulating the wave propagationsuses source positions corresponding to locations of physical receiversat the subterranean formation.
 6. The method of claim 4, wherein the setof spatially discretized partial differential equations include2-dimensional stress-velocity two-way elastic wave equations.
 7. Themethod of claim 1, wherein the wave propagation data is generated with avelocity model associated with the subterranean formation.
 8. The methodof claim 1, wherein the seismic data collected from the plurality ofreceivers includes pressure data or particle velocity data.
 9. Themethod of claim 1, wherein reconstructing the time-reversed wavefielddata is based on reconstructing a target area of the subterraneanformation.
 10. The method of claim 1, wherein the vectorized snapshotmatrix comprises wave propagation data and absorbing layer values.
 11. Acomputing system for seismic event localization, the system comprising:at least one memory and at least one processor configured for:generating a vectorized snapshot matrix representing wave propagationdata at a series of snapshots in time for a subterranean formation;computing a reduced orthonormal column basis matrix based on thevectorized snapshot matrix; constructing a reduced order wavepropagation model based on the reduced orthonormal column basis matrix;receiving seismic data collected from a plurality of receivers at thesubterranean formation; generating a time-domain coefficient matrixbased on back propagation of the received seismic data and the reducedorder wave propagation model; reconstructing time-reversed wavefielddata based on the time-domain coefficient vector; and generating signalsfor outputting wavefield or seismic event location information based onthe time-reversed wavefield data.
 12. The system of claim 11, whereincomputing the reduced orthonormal column basis matrix comprises:singular value decomposition or QR decomposition.
 13. The system ofclaim 11, wherein computing the reduced orthonormal column basis matrixis based on multi-dimensional stress-velocity two-way elastic waveequations:$\frac{\partial{v_{x}(t)}}{\partial t} = {{\frac{1}{\rho}\frac{\partial{\tau_{xx}(t)}}{\partial x}} + {\frac{1}{\rho}\frac{\partial{\tau_{xz}(t)}}{\partial z}} + {f_{v_{x}}(t)}}$$\frac{\partial{v_{z}(t)}}{\partial t} = {{\frac{1}{\rho}\frac{\partial{\tau_{xz}(t)}}{\partial x}} + {\frac{1}{\rho}\frac{\partial{\tau_{zz}(t)}}{\partial z}} + {f_{v_{z}}(t)}}$$\frac{\partial{\tau_{xx}(t)}}{\partial t} = {{\left( {\lambda + {2\mu}} \right)\frac{\partial{v_{z}(t)}}{\partial x}} + {\lambda\frac{\partial{v_{z}(t)}}{\partial z}} + {f_{p}(t)}}$$\frac{\partial{\tau_{zz}(t)}}{\partial t} = {{\left( {\lambda + {2\mu}} \right)\frac{\partial{v_{z}(t)}}{\partial z}} + {\lambda\frac{\partial{v_{x}(t)}}{\partial x}} + {f_{p}(t)}}$${\frac{\partial\tau_{x{z{(t)}}}}{\partial t} = {\mu\left( {\frac{\partial{v_{x}(t)}}{\partial z} + \frac{\partial{v_{z}(t)}}{\partial x}} \right)}},$where τ_(xx) and τ_(zz) are the x and z components of the normal stressfields; τ_(xz) is shear stress field; v_(x) and v_(z) are horizontal andvertical components of particle velocity fields; f_(v) _(x) and f_(v)_(z) are single forces and f_(p) is an external pressure source
 14. Thesystem of claim 11, wherein generating the vectorized snapshot matrixcomprises: simulating wave propagations on a set of spatiallydiscretized partial differential equations; and extracting simulatedwave propagation data at each snapshot in the series of snapshots intime.
 15. The system of claim 14, wherein simulating the wavepropagations uses source positions corresponding to locations ofphysical receivers at the subterranean formation.
 16. The system ofclaim 14, wherein the set of spatially discretized partial differentialequations include 2-dimensional stress-velocity two-way elastic waveequations.
 17. The system of claim 11, wherein the wave propagation datais generated with a velocity model associated with the subterraneanformation.
 18. The system of claim 11, wherein the seismic datacollected from the plurality of receivers includes pressure data orparticle velocity data.
 19. The system of claim 11, whereinreconstructing the time-reversed wavefield data is based onreconstructing a target area of the subterranean formation.
 20. Thesystem of claim 11, wherein the vectorized snapshot matrix compriseswave propagation data and absorbing layer values.